Inextensional wrapping of flat membranes

Abstract

In this paper we are concerned with the wrapping of a flat, thin membrane around a central hub. The folding pattern consists of a series of hill and valley folds, as in a recent proposal by Temple and Oswald for the design of a solar sail. For launch their sail is wrapped around the circular body of the spacecraft, about 4 m in diameter; once in orbit it is deployed to a 276 m diameter disk which can collect enough solar pressure to sail to Mars. The paper begins with a brief survey of related work: it turns out that, since the early 1960's, several people have thought about folding thin sheets in this way, and yet no complete solution or even explanation is available. We present a simple description of the folding technique. Based on the simplifying assumption that the membrane to be folded has zero thickness, we identify some key properties of the folding pattern and hence show how to draw the fold pattern. Then we present a simple way of computing the correct fold pattern for thin membranes. We discuss some alternative fold patterns, including irregular hubs and other variants. In the Appendix we give two fold patterns from which simple demonstration models can be made.